The gluing formula for fhat{Z}-invariants of glued knot complements

Let $K_1$ and $K_2$ be knots with well-defined GM series, and let

$$M=(S^3\setminus\nu(K_2))\cup_\gamma(S^3\setminus\nu(K_1))$$

be the manifold obtained by gluing with $\gamma=\begin{pmatrix}r&a\p&b\end{pmatrix}$. Write the GM series as $F_{K_i}(X,q)=\sum_{m\in\mathbb{Z}+\frac12}f^{m}_{K_i}(q)X^m$. Gluing Formula. The invariant is

$$\hat{Z}_{\alpha}(q)=\sum_{m_1,m_2\in\mathbb{Z}+\frac12}f^{m_1}_{K_1}(q)f^{m_2}_{K_2}(q)q^{-\frac1p(bm_1^2+2m_1m_2+rm_2^2)}\delta^{\mathbb{Z}}\left(\frac{m_1+rm_2}{p}+\frac{\alpha}{p}\right),$$

so long as the right-hand side converges. This is expected beyond the plumbed-knot case whenever the GM series are well-defined; convergence remains part of the assertion.